Pneumatic Performance Study of a High Pressure Ejection Device Based on Real Specific Energy and Specific Enthalpy

Transcription

1 Entropy 14, 16, ; doi:1.339/e Artile OPEN ACCESS entropy ISSN Pneumati Performane Study of a High Pressure Ejetion Devie Based on Real Speifi Energy and Speifi Enthalpy Jie Ren *, Fengbo Yang, Dawei Ma, Guigao Le and Jianlin Zhong Shool of Mehanial Engineering, Nanjing University of Siene and Tehnology, Nanjing 194, Jiangsu, China; s: yangfengbo.ool@163.om (F.Y.); ma-dawei@mail.njust.edu.n (D.M.); leguigao@njust.edu.n (G.L.); njustzhongjianlin@163.om (J.Z.) * Author to whom orrespondene should be addressed; renjie@njust.edu.n; Tel.: Reeived: 8 May 14; in revised form: 1 August 14 / Aepted: 9 August 14 / Published: 3 September 14 Abstrat: In high-pressure dynami thermodynami proesses, the pressure is muh higher than the air ritial pressure, and the temperature an deviate signifiantly from the Boyle temperature. In suh situations, the thermo-physial properties and pneumati performane an t be desribed aurately by the ideal gas law. This paper proposes an approah to evaluate the pneumati performane of a high-pressure air atapult launh system, in whih esidual funtions are used to ompensate the thermal physial property unertainties of aused by real gas effets. Compared with the Nelson-Obert generalized ompressibility harts, the preision of the improved virial equation of state is better than Soave-Redlih-Kwong (S-R-K) and Peng-Robinson (P-R) equations for high pressure air. In this paper, the improved virial equation of state is further used to establish a ompressibility fator database whih is applied to evaluate real gas effets. The speifi residual thermodynami energy and speifi residual enthalpy of the high-pressure air are also derived using the modified orresponding state equation and improved virial equation of state whih are trunated to the third virial oeffiient. The pneumati equations are established on the basis of the derived residual funtions. The omparison of the numerial results shows that the real gas effets are strong, and the pneumati performane analysis indiates that the real dynami thermodynami proess is obviously different from the ideal one. Keywords: thermodynamis; residual funtion; speifi thermodynami energy; speifi enthalpy; high pressure air; ompressibility fator

2 Entropy 14, Introdution Compared to petroleum or eletri systems, high pressure air has the advantages of no pollution, high power density, heapness, reliable performane, reyling use, and being easy to maintain [1,]. It has been applied to industrial automation, robot driving, ompressed air powered vehiles, and even some speial industries suh as aeronautis, astronautis, and weapons design [3]. Typially, the harging and disharging performane and exergy analysis [4] are based on the equations of state. However, in high-pressure pneumati dynami thermodynami proesses, the pressure is muh higher than the air ritial pressure and the temperature an deviate signifiantly from the Boyle temperature. Sine the thermodynami properties based on the ideal gas are not authenti [5], the deviations in the mass and energy balanes in thermodynami proess may not be aeptable. In this paper, the speifi residual thermodynami energy and speifi residual enthalpy will be derived to ompensate for the real gas effet. Extensive studies have been onduted on high pressure pneumati systems [6,7]. In these works, most of the thermodynami property alulations are still based on the ideal gas assumption, although the speifi thermodynami energy and speifi enthalpy of an ideal gas and a real gas an differ onsiderably under high pressure and low temperature onditions. In this study, we will also examine the deviation between the state variables omputed by the real gas equation and ideal gas assumption. Many semi-empirial formulas have been proposed to desribe the properties of real gases, inluding the van der Waals equation [5], Redlih-Kwong (R-K) equation [8], Soave-Redlih-Kwong (S-R-K) equation [9], Benedit-Webb-Rubin (B-W-R) equation [1], and Peng-Robinson (P-R) equation [11,1]. With the development of the orresponding states priniples, these equations are appliable to all kinds of gases within a ertain pressure and temperature range. However, the preision of some of these equations is not satisfatory, or when alulating the thermodynami variables by using high order nonlinear equations, like the S-R-K and P-R equations, it will lead to the extra problem of solving transendental equations, whih appearane is not desirable. The SAFT-type equation of state [13] is aurate enough to alulate thermodynami variables of air, while the mathematial expression of the derived funtions are omplex.. Derivation and Determination of Real Gas Equation of State for High Pressure Air Aording to the orresponding state law, the ompressibility fator whih indiates the deviation of real gas from ideal gas an be obtained from the table of orresponding states [14]: PV m PV m PV r mr Z Z Pr, Tr RT RT Tr (1) The ompressibility fator is a funtion of the orresponding pressure and temperature. The ompressibility fator value of an ideal gas is 1. For most gases, the P-Z urves an be approximately onsidered to be linear when the pressure P <.5P or the temperature T > 5T, and the ompressibility fator Z is nearly 1. However, under high pressure onditions over.5p, or low temperature onditions below 5T, a reliable and simple real gas equation of state should be derived to fit the data. The virial oeffiients whih are basi thermodynami properties represent the non-ideal behavior of real gases. The importane of the virial oeffiients lies in the fat that they are related diretly to the interations between moleules. The seond virial oeffiient represents the deviation behavior from

3 Entropy 14, ideality due to interations between pairs of moleules, the third virial oeffiient gives the effets of interations of moleular triplets, and so on. The fourth and higher virial oeffiients usually ontribute little to the densities of gases and have relatively large unertainties. Therefore, the aurate knowledge of the virial oeffiients is of great signifiane. In order to improve the auray of alulation, most effort has been foused on obtaining the seond [15,16] and third virial oeffiients [17,18]. The volume serial form of the virial equation whih is trunated to the third virial oeffiient an be written as: PVm B C 1 () RT V V After the introdution of the ritial pressure P, the ritial temperature T, aentri fator ω, and the extended orresponding states variable θ [19], the seond and third virial oeffiients of the orresponding state an be expressed as: m m BP B B ( T ) B ( T ) B ( T ) (3) 1 r r r r r r r RT CP C C ( T ) C ( T ) C ( T ) (4) r 1 r r r r r ( RT ) r where B r (T r ), C r (T r ) are obtained by fitting data for small spherial moleules (ω = ); B 1 r (T r ), C 1 r (T r ) are obtained from data for larger, non-spherial, non-polar moleules (ω ); B r (T r ), C r (T r ) are obtained from data for non-hydrogen bonding polar moleules; and T r = T/T. In this paper, the real gas is assumed to be air in hemial equilibrium []. The National Institute of Standards and Tehnology (NIST) provides a basi model of air [1], whih onsists of nitrogen, oxygen, and argon. These are non polar moleules, the B r (T r ), and C r (T r ) of dry air are, and the ritial parameters of air are: T = 13.45K, P = 3.77MPa. The improved formulas of B r (T r ), B 1 r (T r ), C r (T r ), and C 1 r (T r ) given by referenes [ 4] are: Br ( Tr) (5) T T T T r r r r Br( Tr).1744 (6) 3 8 Tr Tr Tr Tr Cr ( Tr).147 T T (7) r r Cr( Tr).676 (8) T T T T r r r r The Peng-Robinson (P-R) equation, Equation (9), is an improvement on the van der Waals equation. It was proposed in 1976 [1,13]: p RT a( T ) v b v( vb) b( vb) (9)

4 Entropy 14, RT.5 where at ( ) r Tr, p r w w , and.8664rt b. p In the general high pressure pneumati system, the pressure an reah 3 MPa, and the temperature range is about 5 K< T <4 K. Compared with the Nelson-Obert s generalized ompressibility hart [1], the P-R equation is more preise than the S-R-K equation in the ompressibility fator alulation of air [1]. In this paper we alulate the ompressibility fator, using improved virial and P-R equations. The results of some feature points are shown in Table 1. The data for air fitted in this paper are provided by the real properties database of National Institute of Standards and Tehnology (NIST) [5], as shown in Table. Table 1. Compressibility fators of air using improved virial equation and P-R equation. Pressure [MPa] Compressibility Fators Z virial P-R Temperature [K] 98 5 Speifi volume [m 3 /kg] Temperature [K] virial P-R virial P-R virial P-R Table. Speifi volume within different temperature and pressure. Pressure [MPa] The error results when alulating the ompressibility fator within the pressure range of.1135 MPa < P < 3 MPa at the temperatures of 4 K, 3 K and 6 K, respetively, are shown as Table 3. From Table 3, it is found that the preision of improved virial equation is better than P-R equation ompared to the NIST database. Compared with the P-R equation, the preision of the improved virial equation is better. The preision an meet the requirements of general engineering omputation. Therefore, the improved virial equation is adopted to alulate ompressibility fator, and in this paper the thermodynami variables will be derived based on the improved virial equation of state.

5 Entropy 14, Table 3. Error of ompressibility fators for air using improved virial equation and P-R equation respetively. Temperature [K] Pressure range [MPa] P-R equation Maximum absolute error Maximum relative error Improved virial equation Maximum Maximum absolute relative error error ~ % % ~ %.74.47% ~ % % Figure 1 shows the relationship between the pressure and density of high pressure air under different temperature onditions using different equations of state. It is found that the improved virial equation shows good agreement with the NIST data. Figure 1. Comparison of real gas equations of state (T = 4 K, T = 3 K, T = 6 K). (a) (b) () 3. Modeling Thermodynami Variables The dynami thermodynami analysis under high pressure onditions suh as mass flow rate, harging and disharging proesses, and exergy analysis in the pneumati system are of partiular interest in many appliations. Therefore, to investigate the thermal behaviors in thermodynami

6 Entropy 14, proesses, whih an be used to predit the gas pressure, temperature and flow rate, it is essential to derive the thermodynami variables on the basis of real gas equation states. In this setion, the real analytial expressions of speifi thermodynami energy and speifi enthalpy will be derived for high pressure air Residual Funtions In the alulation of real thermodynami property variables, the ideal value an be alulated first, then the residual funtion will subtrated from the ideal value. The definition of residual funtion an be expressed as: F F F (1) re where F re denotes the residual of a arbitrary extensive properties or speifi properties, that is, the differene between the properties of ideal gas and real gas, F * denotes the properties of ideal gas, and F denotes the properties of real gas. The differential form of speifi thermodynami energy for real gas is: P du dt [ p T ( ) ] dv (11) v T v We have: u p T p (1) vt T v The residual speifi thermodynami energy for a real gas is obtained by integrating the equation above from v * = (ideal gas state) to v (real gas state) along the isotherm: v p v ( p/ T) ur p T( ) dv T dv T T (13) v The speifi enthalpy for a real gas is defined as: h u pv (14) and the speifi enthalpy for an ideal gas an be expressed by: * * h u RgT Combination of the above three equations will lead to: (15) v ( p/ T) hr T dvpvrgt T (16) v 3. Thermodynami Variables of Ideal Gas The speifi thermodynami energy and speifi enthalpy of ideal gas an be written as: T * * ( ) V (17) T u T dt u

7 Entropy 14, T * * ( ) ( V g) T (18) h T R dt h For ideal gas, isohori heat apaity an be expressed by: R ( 1) T T T T (19) * 3 4 V g where 3.653, , , , and Thermodynami Variables of Real Gas Based on the improved virial equation and substituting Equations (13), (16), (17) and (18) into Equation (1), respetively, the analytial expressions of speifi thermodynami energy and speifi enthalpy for real gas are obtained as follows: T 3 4 u Rg ( 1) T T T T dt u T 3 8 RRgT.31336T.17T.176T.35T + 7 pv m T T T T 1.313T.438T w.15581t + 7 T T T RRT g.6896t.3865t.4956t.1t.18t.394t w pv m T T T T T T T 3 4 h Rg T T T T dt h T 3 8 RRgT.31336T.17T.176T.35T + 7 pv m T T T T 1.313T.438T w.15581t + 7 T T T RRT g.6896t.3865t.4956t.1t.18t.394t w pv m T T T T T T 1 1 RT Br ( Tr) Br( Tr) Br ( Tr) ( RT) Cr ( Tr) Cr( Tr) Cr ( Tr) RT g vmp vm P () (1) 4. Modeling of Pneumati Catapult Figure shows the working priniple of a high pressure pneumati atapult, and Figure 3 is the shemati diagram of the lifting ejetion mehanism whih is a shemati strutural diagram of the three-step piston ylinder in Figure 1 and omponent 6 in Figure 4. There d a is the diameter of the piston rod, d is the diameter of piston, S a is an effetive thrust area of the third stage ylinder, L b represents the seond ylinder, and L represents the third stage ylinder.

8 Entropy 14, Figure. High pressure gas supply system. Value ontrol Cylinder Hyperbari hamber Low Pressure hamber Cylinder Cylinder Cylinder The working priniple of the lifting ejetion system an be desribed as follows: as the launhing proess starts, the ontrolled valve immediately responds and opens, and high-pressure gas flows into the low pressure hamber; the seond stage of the ylinder pushes the piston to move forward, then the first stage of the ylinder pushes the piston to move forward after the seond stage moves to the end; missile moves with lifting beam, then lifting beam ollides with the buffer, and missile flies out of the launh tube. In the ourse of the pneumati ejetion, taking the subsoni and soni flow into aount, mass flow equation an be written as: G k1 k pa 1 k p p p k k k1 x [( ) ( ) ], ( ) 1 RT k1 p p k1 p x g k1 k ( k1) p k1 1 g 1, 1 k( ) p A/ R T ( ) k1 p k 1 () where subsripts 1 and indiate high-pressure hamber and low pressure hamber respetively, μ x is flow orretion fator, A denotes the equivalent ross-setional area of orifie, and k denotes the adiabati index. Aording to mass and energy onservation laws, and the flow equation in the high-pressure hamber and low pressure hamber, the following relations an be established: d ( 1V1) Qm dt d ( 1Vu 1 1) Qmh1 dt RT1 B1 C1 p1 (1 ) Vm1 Vm1 Vm1 dm Qm (3) dt d dl ( mu ) Qmh1 nsp t dt dt dv me n( p pa) st 1.megsin( ) dt RT B C p (1 ) Vm Vm Vm

9 Entropy 14, where m e is the quality of missile, P a denotes the atmospheri pressure, g is the aeleration of gravity, v is the speed of missile, and s t denotes effetive thrust area. Let X 1 =ρ 1, X =T 1, X 3 =m, X 4 =T, X 5 =l, X 6 =v, the losed pneumati equations an be established as: X 1 G / V1 RRT X RRT G 1 X g M g M * * D111 1 D 11 VdT u G h1 h r / V V1 pv m1 X1Vm 1 pv m1 X1V m1 RRT * g RRT g X1 V D11 D 1 pv m1 pv m1 X 3 G * X X 4 RRT g RRT g RRTM g V nsx 5 RT * V dt e D11 D 1 1 h1 h 5 1r G pv m pv m pv mx3 pv m RRTMnsX g 6 RT XRRT * 3 g XRR 3 g T nspx X 3V D1 D pv m pv m pv m pv m X 5 X 6 n( p p) s 1.Mgsin( ) Mgsin( ), p p 1. M ns X 6 M gsin( ), p p 1. ns T.4344T.13T.76366T.6464T.396T D X X X X X X T.31175T.898T.4T.9T.743T D X X X X X X T.17T.176T D11.35T 7 X 4 X4 X T 1.313T.438T.15581T 7 X 4 X4 X T.3865T.4956T.1T.18T.394T D X4 X4 X4 X4 X4 X T.4344T.13T.76366T.6464T.396T D X4 X4 X4 X4 X4 X T.31175T.898T.4T.9T.743T D X4 X4 X4 X4 X4 X4 (4)

10 Entropy 14, Figure 3. Three-step piston ylinder simplified diagram. L Air intake S a Buffer d L b d a L a Load Figure 4. Lifting ejetion mehanism sketh folding wing, -lifting beam, 3-lifting pole, 4-missile, 5-gas inlet, 6-three-step piston ylinder, 7-exhaust orifie, 8-launh tube. 5. Simulation Analysis of the Pneumati Ejetion The system parameters of above mathematial model and thermo-physial parameters are as shown in Table 4. The five-step four-order Runge-Kutta method is used to alulate the pneumati ejetion proess based on ideal gas equation and real gas equation respetively. The basi idea of five-step four-order Runge-Kutta method is expressed as: ki1 fi( tj, X1j,..., X6j) t tk11 tk61 ki fi tj, X1j,..., X6 j t tk1 tk6 ki3 fi( tj, X1j,..., X6 j ), i 1,,..., 6 ki4 fi( tj t, X1 j tk13,..., X6 j tk63 ) ki 1 ki ki3 ki4 Xi, j1 Xi, jt where t indiates time step, and j represents the urrent time step. (5)

11 Entropy 14, Table 4. System parameters and thermal physial parameters of dry air. Parameters Values Universal gas onstant R [J/mol K] Critial temperature T [K] Critial pressure P [MPa] 3.77 Aentri Fator.31 Gas onstant of air R g [J/kg K] 87. Molar mass M [g/mol] 8.97 Flow orretion fator μ x.95 Cross-setional area valve ontrol A [m ].13 Adiabati index k 1.4 Initial gas density of 1 ρ 1 [kg/m 3 ] 36 Initial gas temperature of 1 T 1 [K] 3 Volume of 1 V 1 [m 3 ] 1.8 Initial gas mass of m [kg] 6 Initial gas temperature of T [K] 3 The number three piston ylinder n 4 Effetive thrust area of ylinder S [m ].35 Mass of missile M [kg] 4, Launh angle [deg] 9 Initial volume of V [m 3 ] Comparative Analysis of Dynami Thermodynami Proesses Figures 5 9 show the dynami variation omparison of the main thermodynami variables in the ejetion proess. When air flows from high pressure hamber to low pressure hamber in the throttling proess, the temperature will hange with the pressure drop. The throttling proess is assumed to be an isenthalpi proess. From Figure 5, we an see that the high pressure air flows into the low pressure hamber, the temperature in the high pressure hamber dereases all the time. In the initial stage, the temperature of the low-pressure hamber inreases immediately, and then dereases gradually when the missile moves upward. Also, from Figure 5, we an see that the real temperature in the high hamber dereases faster than the ideal temperature before. s. This an be explained from the following two aspets: firstly, Figure 9 shows that the gas mass based on real gas and ideal gas flowing into the low pressure hamber are nearly equal; seondly, Figure 7 shows that the residual enthalpy in high pressure hamber is positive, the real enthalpy of high pressure air is obviously less than ideal one. As a result, the real temperature in the high hamber dereases faster before. s. From Figure 5, we know the temperature in the low pressure hamber inreases immediately, and then dereases slowly when the missile moves upward. It shows that the real temperature of gas in the low pressure is always lower than ideal one, whih indiates that the real gas effets deelerates the temperature inreasing rates in the low pressure hamber in the early stage, and aelerates the rate of derease of the temperature with the missile moving upward. Figure 6 shows the pressure variations in the high pressure hamber and low pressure hamber. We an see that the hyperbari always deflates, and the pressure ontinues to deay. High pressure air flows into low pressure hamber that the pressure in the low pressure hamber inreases before the missile

12 Entropy 14, starts to move, and dereases as the missile moves upward. The real pressure deay rate is greater than the ideal deay rate. This is due to the following fats: firstly, from the analysis above, we know that the real temperature in the high hamber dereases faster than the ideal one; seondly, the real gas mass flowing out of high pressure hamber is nearly the same as the ideal mass. The first fator is the dominant fator whih makes the phenomenon happened. The ideal value of temperature in the hyperbari pressure hamber is muh greater than the atual value whih is similar to low pressure hamber. Figure 5. Temperature ontrast urve. Figure 6. Pressure ontrast urve. From Figure 7, we an see that the speifi enthalpy in the hyperbari hamber inreases first and then dereases slightly, while the speifi enthalpy in the low pressure hamber rises rapidly, and then inreases slowly. When the pressure getting higher, the distane between moleules beomes smaller and the intermoleular interation beomes stronger, as a result, the speifi enthalpy gets larger and it deviates more from the ideal gas state. While the temperature inreases, the situation is totally different: longer distane between moleules makes the intermoleular fores smaller, so the speifi enthalpy gets smaller and it is more lose to the ideal gas state. For the high pressure hamber, with gas flowing into low pressure hamber, pressure deays, and temperature dereases rapidly. Figure 7 shows the speifi enthalpy in the high pressure hamber drops slightly, whih indiates that the pressure drop is the main fator. For the low pressure hamber, the pressure and temperature inrease first, and derease with the gas pushing the missile upwards. Figure 7 shows the speifi enthalpy in the low pressure hamber inreases rapidly first and then inreases slightly, whih indiates that the pressure inrease is the main fator in the early stage, and the temperature derease is the main fator in the later round.

13 Entropy 14, Figure 7. The residual enthalpy urve. Figure 8 shows the dynami variation of pressure ompression fators in the hyperbari hamber and low pressure hamber. The maximum ompression fator reahes and 1.486, respetively. From Equation (1) we an see that the ompressibility fator is determined by temperature and pressure. As shown in Figure 5 and Figure 8, the ompressibility fator variation law is similar to the pressure one whih indiates that the pressure dominates the hanging regulation of the ompression fator. Figure 8. The ompressibility fator urve. Figure 9 shows the mass flow rate delines rapidly at first, and then rises slowly. It is also notied in Figure 9 that the real gas effet aelerates the rate of derease of the mass flow in the early stage, and deelerates the rates of inrease in the later period. Figure 9. The mass flow rate urve.

14 Entropy 14, Ejetion Performane Evaluation Variables With the missile overload onsistent with the pressure in the low pressure hamber, agreement between the aeleration of missile and pressure in the low pressure hamber are fairly satisfied, regardless of whether the orrespondene relationship is based on a real gas or based on an ideal gas, just as shown in Figures 5 and 9. The missile speed is a linear funtion of aeleration, and the missile stroke is a quadrati funtion of aeleration, while the ejetion time is less than 1 s. It is shown that the veloity of missile based on an ideal gas is signifiantly greater than that based on a real gas, and missile stroke based on an ideal gas is slightly larger than that based on a real gas shown in Figures 1 1. Figure 1. The aeleration urves. Figure 11. The veloity urve. Figure 1. The displaement urve.

15 Entropy 14, Conlusions In this paper, the improved virial equation of state is used to desribe the thermodynami properties of high pressure air by fitting the NIST data. The ompressibility fator is utilized to evaluate the preision of the equation of state. Compared with the NIST data, the ompressibility fator value obtained from the improved virial equation has a maximum error of 1.33%,.47% and 4.3% within the pressure ranges of.1135 MPa < P < 3 MPa at the temperatures of 4 K, 3 K and 6 K, respetively, and the preision of the improved virial equation of state is better than that of the existing P-R and S-R-K equations. Also, the analytial expression for thermodynami variables, suh as the speifi residual thermodynami energy and speifi residual enthalpy are presented to ompensate the real gas effets, based on the improved virial equation of state. The study on dynami thermodynami analyses, mass flow rate, harging and disharging proesses, and exergy analysis are of partiular importane in high pressure air appliations. In addition, based on the real thermodynami variables, the internal ballistis mathematial model for a pneumati ejetion system is established, with the real gas effets onsidered. Numerial simulations are also performed. The detailed dynami thermodynami proesses for disharging proesses in the hyperbari hamber and harging proesses in the low pressure hamber are analyzed. The omparison of the numerial results indiate that the value of residual enthalpy is high, the state of the working fluid deviates from the ideal gas, and the ompressibility of working fluid is strong, as the ompression fator reahes The real gas effets aelerate the pressure and temperature rates of derease in the hyperbari pressure hamber, and deelerate the rates of inrease in the low pressure hamber. Aknowledgments The authors aknowledge the support from the National Natural Siene Foundation of China (Grant No ) and the Natural Siene Foundation of Jiangsu Provine, China (Grant No. BK13761). Authors Contributions Dawei Ma oneived of the ideal that we an apply high pressure ejetion devie to weapon sope in line with the development diretion for new onept weapons. His main ontribution is the problem motivation. Jie Ren established the struture sheme of the high pressure pneumati atapult, and developed the method to evaluate the real gas effets, by establishing ompression fator library. Jie Ren also dedued the losed pneumati equations to analyse the real dynami thermodynami proess, by presenting the analytial expressions of speifi residual thermodynami energy and speifi residual enthalpy of the high-pressure air based on real gas state equation. Fengbo Yang developed the researh program to study the real gas effets, by using five-step four-order Runge-Kutta method to solve the losed pneumati equations and alulated the ompression fator based on P-R equation and improved virial equation. The ontribution of Jianlin Zhong was to dedue the losed pneumati equations based on ideal gas, inluding the mass and energy onservation laws. Guigao Le was responsible for the omparative analysis of the pneumati atapult performane based on real gas state equation and ideal

16 Entropy 14, gas state equation, inluding the analysis of dynami thermodynami proesses and artile polish. All authors have read and approved the final manusript. Nomenlature Z ompressibility fator, dimensionless h speifi enthalpy, J/kg P pressure, MPa R g gas onstant, J/(kg K) T temperature, K s speifi entropy, J/(kg K) V m molar volume, m 3 /mol α, β, γ, δ, ε onstants R universal gas onstant, J/(mol K) ρ density, (kg/m 3 ) B the seond virial offiient, dimensionless n polytropi exponent, dimensionless C the third virial offiient, dimensionless Subsripts aentri fator, dimensionless r orresponding value Θ extended orresponding states variables, dimensionless ritial value u speifi thermodynami energy, J/kg re residual value C v isohori heat apaity, J/(kg K) * ideal gas v speifi volume, m 3 /kg referene state Conflit of Interest The authors delare no onflit of interest. Referenes 1. Luo, Y.; Wang, X. Exergy analysis on throttle redution effiieny based on real gas equations. Energy 1, 35, Zhu, J.; Lei, J.; Huang, Z.; Fu, T. Pneumati position servo ontrol system based on grey relational ompensation ontrol. Chin. J. Meh. Eng. 1, 48, (In Chinese) 3. Wang, X.; Luo, Y.; Xu, Z. Study of polytropi exponent based on high pressure swithing expansion redution. J. Therm. Si. 11, 5, Yang, J.L.; Ma, Y.T.; Li, M.X.; Guan, H.Q. Exergy analysis of transritial arbon diodide refrigeration yle with an expander. Energy 5, 3, Chen, Z. Advaned Engineering Thermodynamis; Higher Eduation Press: Beijing, China, Jia, G.; Wang, X.; Liu, H.; Wu, G. Study on pressure redution and power property of high pressure pneumati system on the ompressed air powered vehile. China Meh. Eng. 4, 15, (In Chinese) 7. Yang, G.; Guo, J.; Li, B. Dynami simulation investigation of a novel high-pressure pneumati proportional ontrol valve. China Meh. Eng. 7, 18, (In Chinese) 8. Privat, R.; Privat, Y.; Jaubert, J.-N. Can ubi equations of state be reast in the virial form? Fluid Phase Equilibr. 9, 8, Jaubert, J.-N.; Privat, R. Relationship between the binary interation parameters (k ij ) of the Peng Robinson and those of the Soave Redlih Kwong equations of state: Appliation to the definition of the PRSRK model. Fluid Phase Equilibr. 1, 95, Salise, O.H. On the phase equilibrium stokmayer fluids. Fluid Phase Equilibr. 7, 53,

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